\beginsection 1.2

Prove $3+11+\cdots+(8n-5)=4n^2-n$ for all natural numbers $n$.

\medskip

Induction Step 1: Show that $P_1$ is true.
$$P_1=(8\cdot1-5)=3$$
Induction Step 2: Show that $P_n+(8(n+1)-5)=P_{n+1}$.
$$\eqalign{
P_n+(8(n+1)-5)&=4n^2-n+8n+3\cr
&=4n^2+7n+3\cr
\cr
P_{n+1}&=4(n+1)^2-(n+1)\cr
&=4(n^2+2n+1)-n-1\cr
&=4n^2+8n+4-n-1\cr
&=4n^2+7n+3\cr
}$$
